For each sinusoidal component of a signal, we need to determine its
frequency, amplitude, and phase (when needed). As a starting point,
consider the windowed complex sinusoid with complex amplitude
and frequency
:

(11.20)

As discussed in Chapter 5, the transform (DTFT) of this
windowed signal is the convolution of a frequency domain delta
function at
[
], and the
transform of the window function,
, resulting in a shifted
version of the window transform
.
Assuming
is odd, we can show this as follows:

Hence,

At
, we have

If we scale the window to have a dc gain of 1, then the peak magnitude
equals the amplitude of the sinusoid, i.e.,
, as shown in Fig.10.8.

Figure:
Schematic diagram of a window
transform amplitude-scaled by
and frequency-shifted by
.

If we use a zero-phase (even) window, the phase at the peak equals the
phase of the sinusoid, i.e.,
.